Competition problem — Number theory

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Number theory Difficulty 5.8 AIME, harder Prove it

26. Prove: The indeterminate equation $x^{2}-8 y^{4}=1$ has no other positive integer solutions except $x=3, y=1$ .

Solution

26. $(x-1)(x+1)=8 y^{4}$ , there must be (i) $x-1=2 u^{4}, x+1=2^{4 r+2} v^{4}$ ; or (ii) $x-1=$ $2^{4 r+2} u^{4}, x+1=2 v^{4}, 2 \nmid u v,(u, v)=1$ . Similar to the discussion in problem 24, and using problem 25.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.